Extracting filtered signal statistics of continuously measured quantum systems

Abstract

The joint state of a continuously monitored quantum system and the classical filtered measurement record has recently been shown to be described by a quantum Fokker-Planck master equation [Phys. Rev. Lett. 129, 050401 (2022)]. We present a deterministic approach to compute the steady state of the system and detector. The method is shown to become particularly efficient in the absence of feedback, which we exploit to develop a perturbative approach valid for weak feedback. We show that through this method we can extract the full counting statistics of the signal, the quantum-classical mutual information between system and signal, as well as the Fisher information of the signal, which can be used for sensing applications. Our results are illustrated with both single-qubit models, as well as the spin chains governed by the one-dimensional transverse field Ising model or the Lipkin-Meshkov-Glick model.

Figures (9)

• Figure 1: Measurement outcomes $P(D)$ solving charge resolved master equation (solid red line) and collating 5000 stochastic trajectories (orange histograms) for $\Omega=1$, $\lambda=0.5$, $\gamma=2$ and $A=\sigma_z$ at different times (a) $t=\pi/8$ (b) $t=\pi/4$ (c) $t=3 \pi/8$ and (d) $t=\pi/2$.
• Figure 2: Steady state distribution of detector outcomes $P(D)$ for increasing values of measurement strength $\lambda=\{0.5,1,1.5,2,2.5\}$ for darker shaded lines with measurement bandwidth (a) $\gamma=0.5$ and (b) $\gamma=1$.
• Figure 3: System detector correlations for the driven qubit. (a) Evolution of $\rm{Cov}(\sigma_z)$ with time for $\lambda=\{0.1,0.5,1.0,2.0\}$ (red solid, blue dashed, green dotted and black dot dashed lines). Steady state values are shown by faded gray lines. (b) Steady state mutual information $\mathcal{I}$ against measurement strength $\lambda$ for different bandwidths $\gamma=\{0.5,1,1.5\}$(red solid, blue dashed and green dotted lines). (c) Steady state covariance for $\sigma_y$ and (d) $\sigma_z$. Other parameter values: $\Omega=1$, $\gamma=0.5$.
• Figure 4: Two time correlation function for the measurement current $C(\tau)$ with parameters $\Omega=1$ and $\gamma=\{0.6,1,1.4\}$(solid red, dashed blue line and dotted green line) for measurement strengths (a) $\lambda=1$ and (b) $\lambda=1.9$.
• Figure 5: Fisher information $F_\mu$ of the steady state distribution $P_\mu(D)$ against measurement strength $\lambda$ for different bandwidths $\gamma=\{0.8,1,1.2,1.4,1.6\}$ (solid red, dashed blue, dotted green, dashed black and short dashed purple lines). Each panel shows results for different values of the reference Rabi frequency (a) $\Omega=0.2$, (b) $\Omega=0.4$, (c) $\Omega=0.6$, and (d) $\Omega=0.8$.